Bradford City supporters knew their trip to Wembley was an experience to enjoy, regardless of the result. Photograph: Tom Jenkins for the Guardian

At least Bradford City do not have to worry now about the cost of all those trips abroad. One suspects they might have found a way round it but for all the glories that would have come with being a League Two club playing in Europe, one of the charming facts about life among the hoi polloi is that they were not entirely sure, logistically, whether they would be able to afford it.

The other consolation is that they have still delivered one of the great football stories of modern times. However raw it feels now, whatever the sense that they have stopped just short of a football miracle, they can still reflect on a once-in-a-lifetime story.

Then, of course, they will have to shake their heads clear. It is Dagenham & Redbridge at Valley Parade on Wednesday. Bootham Crescent, York, on Saturday.

Perhaps, ultimately, the occasion was simply too much. They had looked like sightseers before the game, holding up their camera-phones, taking turns to snap one another in their crisp suits, straight from the peg of an Ilkley tailor. Maybe the emotion of the past few weeks caught up with them.

Whatever the truth, it was the first time in this epic run when they had looked what they are: the 11th-placed team in what football people of a certain generation will always know as the Fourth Division.

A club in this position are not accustomed to the kind of opponent who can nutmeg someone from five yards as Pablo Hernández did to Gary Jones for Michu to make it 2-0. That pass epitomised the class of Swansea. Michu's finish was not too shoddy either: very little back-lift, through Carl McHugh's legs and into the bottom corner.

Wembley will not see many double-nutmegs better than that and it was then, irrespective of the three goals that followed, that everyone knew the fairytale had run its course. From that moment, it was always going to be a procession, an exercise in damage-limitation.

The team assembled for £7,500 – not forgetting the friendly against Guiseley as part of the deal for James Hanson – had used up all their powers of the extraordinary. Hanson, the former shelf-stacker from Eccleshill Co-op, had played like Tommy Lawton against the terrorised defences of Arsenal and Aston Villa. Now he just looked like a run-of-the-mill striker trying to get on the end of scraps.

Only the most flint-hearted observer would not feel any compassion for Matt Duke, one of the heroes of Bradford's cup run, but that is what can happen when two sides are this mismatched. There were bound to be casualties.

Phil Parkinson's side did not manage a shot on target until the 88th minute. Their first corner had arrived two minutes earlier. Swansea, brilliant in every respect, were shrewd enough to give away only one free-kick in their own half, acutely aware that Bradford's strength came from set pieces.

The possession statistics told us that the underdogs had 22.4% of the play. When Duke brought down Jonathan De Guzman for the penalty it was the first foul Bradford had conceded – mainly because they just could not get near enough. Bradford put together 278 passes, Swansea 986.

"It was tough," Parkinson said. "It could have been seven or eight. As you probably know, one of the Swansea players' wages is probably more than our budget for the whole season. That probably sums up the gulf between the sides. And what they've done to us they've done to Premier League sides, too. We've had them watched a lot of times and every time the report back told us this is a side with not many weaknesses. We could have done a lot of things better."

That does not mean defeat comes without dignity. "This final will remain in history," Michael Laudrup, Swansea's manager, said. "A small part because of us, a large part because of Bradford." His players had given the beaten side a guard of honour. "A touch of class," Parkinson said.

Their fans were tremendous, too. After Nathan Dyer had put Swansea into the lead, the Bradford end was on its feet, imploring their team to find new strength. At 70 minutes, when the flags started to flutter again, Swansea's fans applauded their rivals for this show of colour and passion.

These were supporters who were determined their big day should not pass quietly. Their team had not just beaten Arsenal, Aston Villa and Wigan Athletic to get here but Watford from two divisions up and Notts County from the tier above. They were absolutely determined to be heard. "That was probably the highlight for me," Parkinson said. "The way the fans stayed with us. They reminded us that what we've done to get here is unprecedented."

One had come from New Zealand, another from Tahiti but the vast majority had flooded down the motorway. A little aside, but how impressive it was, too, to see the rewards of all the work the club have done trying to embrace the city's Asian community.

Back in Yorkshire, the statue of Sir Titus Salt wore a Bradford scarf. One car on the M1 had undergone a claret and amber makeover. Another was emblazoned with the words "C'mon City". When Manchester United, Chelsea, Liverpool and the other usual suspects visit Wembley it can feel sometimes as if the glamour of old is no longer there. It is when the newcomers arrive that you are reminded what this place means to football people. However corny it may sound, it can reinvigorate your affection for the old place.

For many, it was an emotional day, too. What happened on 11 May 1985, when Bradford played Lincoln City and the wooden stand at Valley Parade became an inferno, is often described as the forgotten disaster. Not here. The giant banner that was passed overhead said: "Always Remembered". Bradford's players wore tracksuits as a tribute to those who died, bearing the number "56" and the words "Always With Us".

Martin Fletcher was 12 when he was dragged from the fire. His brother, Andrew, 11, their father, John, 34, uncle Peter, 32, and grandfather Eddie, 63, did not make it. "I ordered a shirt the day after we reached Wembley with 'Fletcher 56' on the back, as I wanted them all with me in spirit," he said. "For the club to issue the 56 track-jacket showed me I was not alone. It was an unbelievable gesture. They've even made a fan out of my mum, who was wearing one as she watched on television."

Martin had returned to Valley Parade when it reopened and went to games until 2000. This season, starting with the quarter-final against Arsenal, was the first time in 13 years he has been tempted back. "After we beat Aston Villa, I realised these were my people, the people who braved a blazing stand to throw me from the front of it.

"Bradford is where I came from, where my family came from, the place that defined me, and I remembered that where I came from you never forget your roots. I do wonder if I will watch Bradford City again after today. It seems the perfect opportunity to let my final memory be a happy one and bid a final fond farewell to my loved ones."

At most Wembley finals these days the losing end is deserted within a few minutes of the final whistle. Bradford's supporters decided en masse to stick around. "It's a shame we couldn't finish the fairytale," as another fan, Phil Jordan, 30, put it. "But reaching Wembley has given us enough memories for a lifetime."